Where is the exit point?

We consider the exit problem for Kramers' model of noise-activated escape from a potential well. In this singular perturbation problem the small parameter ε is the noise strength (temperature measured in units of barrier height). The stochastic dynamics of the escaping trajectories, conditioned on not returning to a given critical energy contour, are studied analytically and numerically. The distribution of exit points on the boundary of the domain of attraction of the stable equilibrium point in the phase plane is shown to be spread on the separatrix away from the saddle point. In this problem large deviations theory fails to predict the distribution of the exit point for finite noise. It is shown, both by a numerical solution of the conditioned dynamics and analytically, that most of the probability is located at a distance O(ε) from the saddle point and vanishes at the saddle point.